Nuprl Lemma : fill-type-up-0
ā[Gamma:jā¢]. ā[A:{Gamma.š ⢠_}]. ā[cA:Gamma.š ⢠CompOp(A)]. ā[u:{Gamma ⢠_:(A)[0(š)]}].
((app(fill-type-up(Gamma;A;cA); (u)p))[0(š)] = u ā {Gamma ⢠_:(A)[0(š)]})
Proof
Definitions occuring in Statement :
fill-type-up: fill-type-up(Gamma;A;cA)
,
composition-op: Gamma ⢠CompOp(A)
,
interval-0: 0(š)
,
interval-type: š
,
cubical-app: app(w; u)
,
csm-id-adjoin: [u]
,
cc-fst: p
,
cube-context-adjoin: X.A
,
csm-ap-term: (t)s
,
cubical-term: {X ⢠_:A}
,
csm-ap-type: (AF)s
,
cubical-type: {X ⢠_}
,
cubical_set: CubicalSet
,
uall: ā[x:A]. B[x]
,
equal: s = t ā T
Definitions unfolded in proof :
uall: ā[x:A]. B[x]
,
member: t ā T
,
subtype_rel: A ār B
,
all: āx:A. B[x]
,
implies: P
ā Q
,
uimplies: b supposing a
,
csm-ap-term: (t)s
,
fill-type-up: fill-type-up(Gamma;A;cA)
,
cubical-app: app(w; u)
,
cubical-lambda: (λb)
,
swap-interval: swap-interval(G;A)
,
csm-swap: csm-swap(G;A;B)
,
cubical-type: {X ⢠_}
,
interval-0: 0(š)
,
csm-id-adjoin: [u]
,
csm-ap: (s)x
,
cc-fst: p
,
cc-adjoin-cube: (v;u)
,
cc-snd: q
,
csm-ap-type: (AF)s
,
csm+: tau+
,
csm-adjoin: (s;u)
,
csm-id: 1(X)
,
csm-comp: G o F
,
pi1: fst(t)
,
pi2: snd(t)
,
compose: f o g
,
cube-context-adjoin: X.A
,
cubical-term-at: u(a)
,
interval-type: š
,
constant-cubical-type: (X)
,
squash: āT
,
true: True
,
prop: ā
Lemmas referenced :
csm-ap-type_wf,
cube-context-adjoin_wf,
interval-type_wf,
cubical_set_cumulativity-i-j,
csm-id-adjoin_wf-interval-0,
cc-fst_wf,
cubical-term_wf,
cubical-type-cumulativity2,
composition-op_wf,
cubical-type_wf,
cubical_set_wf,
fill-type-up_wf,
cubical-app_wf_fun,
csm-ap-term_wf,
I_cube_wf,
fset_wf,
nat_wf,
cubical-term-equal,
cube_set_restriction_pair_lemma,
interval-type-ap-morph,
dM-lift-0-sq,
cubical-term-at_wf,
filling_term_0,
face-0_wf,
csm+_wf_interval,
csm-face-0,
context-subset-term-0,
constrained-cubical-term-0,
csm+_wf,
csm-id-adjoin_wf,
csm-interval-type,
interval-0_wf,
cc-snd_wf,
csm-composition_wf,
cc-adjoin-cube_wf,
cube-set-restriction_wf,
nh-id_wf,
subtype_rel-equal,
cubical-type-at_wf,
cube-set-restriction-id,
csm-ap-term-at,
cubical_type_at_pair_lemma,
squash_wf,
true_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
instantiate,
hypothesis,
applyEquality,
because_Cache,
sqequalRule,
universeIsType,
inhabitedIsType,
lambdaFormation_alt,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
dependent_functionElimination,
independent_functionElimination,
functionExtensionality,
independent_isectElimination,
setElimination,
rename,
productElimination,
Error :memTop,
applyLambdaEquality,
lambdaEquality_alt,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
hyp_replacement
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma.\mBbbI{} \mvdash{} \_\}]. \mforall{}[cA:Gamma.\mBbbI{} \mvdash{} CompOp(A)]. \mforall{}[u:\{Gamma \mvdash{} \_:(A)[0(\mBbbI{})]\}].
((app(fill-type-up(Gamma;A;cA); (u)p))[0(\mBbbI{})] = u)
Date html generated:
2020_05_20-PM-04_54_50
Last ObjectModification:
2020_04_13-PM-02_56_43
Theory : cubical!type!theory
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